*All posts from*

__olympiads.hbcse.tifr.res.in__

# olympiads.hbcse.tifr.res.in Regional Mathematical Olympiad-2016 : Homi Bhabha Centre For Science Education

** Name of the Centre **: Homi Bhabha Centre For Science Education

**: Regional Mathematical Olympiad-2016**

__Name Of The Exam__**: Mathematical**

__Name Of The Subject__**: Previous Question Papers**

__Document type__**: 2016**

__Year__**: http://olympiads.hbcse.tifr.res.in/how-to-prepare/past-papers/**

__Website__**:**

__Download Model/Sample Question Paper__**CRMO 2016 Paper 1**: https://www.pdfquestion.in/uploads/13118-crmo-16paper1.pdf

**CRMO 2016 Paper 2**: https://www.pdfquestion.in/uploads/13118-crmo-16paper2.pdf

**CRMO 2016 Paper 3 :**https://www.pdfquestion.in/uploads/13118-crmo-16paper3.pdf

## Regional Mathematical Olympiad Question Paper :

**Time: 3 hours
Maximum marks: 100.**

**Instructions :**

** Calculators (in any form) and protractors are not allowed.

** Rulers and compasses are allowed.

Related: Homi Bhabha Centre For Science Education Pre-Regional Mathematical Olympiad Previous Question Paper : www.pdfquestion.in/9136.html

** Answer all the questions.

** All questions carry equal marks.

** Answer to each question should start on a new page. Clearly indicate the question number.

1. Let ABC be a right-angled triangle with \B = 90. Let I be the in centre of ABC. Draw a line perpendicular to AI at I. Let it intersect the line CB at D. Prove that CI is perpendicular to AD and prove that ID = p b(b = a) where BC = a and CA = b.

2. For any natural number n, expressed in base 10, let S(n) denote the sum of all digits of n. Find all natural numbers n such that n = 2S(n)2.

3. Find the number of all 6-digit natural numbers having exactly three odd digits and three even digits. 5. Let ABC be a triangle with centroid G. Let the circum circle of triangle AGB intersect the line BC in X different from B; and the circum circle of triangle AGC intersect the line BC in Y different from C. Prove that G is the centroid of triangle AXY .

4. . Let ha1; a2; a3; : : :i be a strictly increasing sequence of positive integers in an arithmetic progression. Prove that there is an innate sub sequence of the given sequence whose terms are in a geometric progression.

**Paper – II :**

1. For any natural number n, expressed in base 10, let S(n) denote the sum of all digits of n. Find all natural numbers n such that n3 = 8S(n)3+6nS(n)+1.

2. How many 6-digit natural numbers containing only the digits 1,2,3 are there in which 3 occurs exactly twice and the number is divisible by 9?

3. Show that the innite arithmetic progression h (1,4,7,10,….) has infinitely many 3-term sub sequences in harmonic progression such that for any two such triples (a1; a2; a3) and (b1; b2; b3) in harmonic progression, one has (a1/b1 = a2/b2)

**Enrollment of Mathematical Olympiad :**

** The Mathematical Olympiad programme follows a four-stage selection procedure beginning with PRMO.

** Before enrolling for PRMO, check that you satisfy the eligibility criteria.

**Enrollment for PRMO :**

** The PRMO will be a machine-correctable test of 30 questions. Each question has an answer which is a number with one or two digits. Sample PRMO questions and Sample OMR sheet showing marked answers can be downloaded from here.

** The PRMO exam will be organized this year by IAPT (the Indian Association of Physics Teachers), the same association that also organizes the National Standards Examination, which is the first step for participation in the International Physics, Chemistry, Biology, Astronomy and Junior Science olympiads. The website for PRMO is available at iapt.org.in/

** A link to a portal is available at the IAPT webpage where schools can register as centres; this portal will be open from May 20 to June 20, 2017. Any school with at least 5 registrations can register on the portal as a “registered centre”.

** However, the list of approved exam centres will be issued by IAPT. Students who register through a given school which is a registered centre may be re-assigned to another nearby exam centre.

** All Kendriya Vidyalaya, Jawahar Navodaya Vidyalaya and Atomic Energy Central Schools may register as registered centres regardless of the number of students.

** The list of registered centres will be available on the IAPT website from June 20, 2017.

** Students can approach one of the approved registered centres and through the centre register for the PRMO exam between June 20 and July 25, 2017 for a fee of Rs. 200. Kendriya Vidyalaya Sangathan has made its schools available as centres free of cost; for KV students there is a provision to register with payment of a fee of Rs. 100.

** Queries regarding PRMO may be sent by email to prmo@hbcse.tifr.res.in. Queries will not be replied to individually, but via the FAQ section on this website.